A numerical scale for non locally connected planar continua

TL;DR

This paper introduces a numerical scale to measure the degree of non-local connectedness in planar continua, providing a quantitative tool for classifying and analyzing complex topological structures.

Contribution

It defines a new numerical scale for non-local connectedness and explores its properties, including applications to classical fractals like the Mandelbrot set.

Findings

The scale is zero for locally connected continua.

The scale is one for the topologist's sine curve.

The scale is infinite for indecomposable continua.

Abstract

We introduce a numerical scale to quantify to which extent a planar continuum is not locally connected. For a locally connected continuum, the numerical scale is zero; for a continuum like the topologist's sine curve, the scale is one; for an indecomposable continuum, it is infinite. Among others, we shall pose a new problem that may be of some interest: can we estimate the scale from above for the Mandelbrot set $\mathcal{M}$ ?